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  • Open Daily: 10am - 10pm
    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Computation of Invariant Measures with Dimension Reduction Methods

Computation of Invariant Measures with Dimension Reduction Methods

Kemper, Jens

Paperback

General Mathematics

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ISBN10: 3832524525
ISBN13: 9783832524524
Publisher: Logos Verlag Berlin
Published: Apr 30 2010
Pages: 158
Weight: 1.00
Height: 0.40 Width: 5.68 Depth: 8.04
Language: English
In recent years, Dellnitz, Junge and co-workers developed a subdivision algorithm for the approximation of invariant measures in discrete dynamical systems based on the so-called Ulam's approach. In high dimensions, this adaptive invariant measure (AIM) algorithm suffers from the curse of dimension even when the support of the system's invariant measure is known to be low-dimensional. In our thesis we develop algorithms facing this problem by combining the subdivision technique with proper orthogonal decomposition (POD) as a model reduction method. We derive explicit error bounds concerning the long-time behavior of POD solutions, propose a discrete version of the Prohorov metric as a proper distance notion for discrete measures computed by the algorithms, and analytically compare the approximation processes of the AIM algorithm and the POD-based algorithms. A marginal-like representation of discrete measures is proposed in order to visualize the numerical experiments. The algorithms are applied to finite element discretizations of the Chafee-Infante problem in order to show the power of our approach.

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